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Symmetric products of the Euclidean spaces and the spheres

Naotsugu Chinen (2015)

Commentationes Mathematicae Universitatis Carolinae

By F n ( X ) , n ≥ 1 , we denote the n -th symmetric product of a metric space ( X , d ) as the space of the non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric d H . In this paper we shall describe that every isometry from the n -th symmetric product F n ( X ) into itself is induced by some isometry from X into itself, where X is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the n -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and...

The area formula for W 1 , n -mappings

Jan Malý (1994)

Commentationes Mathematicae Universitatis Carolinae

Let f be a mapping in the Sobolev space W 1 , n ( Ω , 𝐑 n ) . Then the change of variables, or area formula holds for f provided removing from counting into the multiplicity function the set where f is not approximately Hölder continuous. This exceptional set has Hausdorff dimension zero.

The boundary absolute continuity of quasiconformal mappings (II).

Juha Heinonen (1996)

Revista Matemática Iberoamericana

In this paper a quite complete picture is given of the absolute continuity on the boundary of a quasiconformal map B3 → D, where B3 is the unit 3-ball and D is a Jordan domain in R3 with boundary 2-rectifiable in the sense of geometric measure theory. Moreover, examples are constructed, for each n ≥ 3, showing that quasiconformal maps from the unit n-ball onto Jordan domains with boundary (n - 1)-rectifiable need not have absolutely continuous boundary values.

Currently displaying 161 – 180 of 245